[6tisch-security] (simplified explanation of formulae) Re: Latency aspects of TSCH

Hi Malisa:

I had a closer look at the slides you presented at the 6TiSCH security 
call this morning, Feb 17, 2015.

Slide 3 presents time latency numbers in case of a random TSCH schedule 
with C cells per slotframe, where the slotframe length is L time slots. 
I was trying to come up with an easy explanation for your time latency 
figure of ~ 1/(C+1)  the slotframe length between receiving a frame and 
sending a response in the next available dedicated "sending cell".

It turns out one can do this without any nontrivial math. See below.

Assumptions:
-w.l.o.g., receipt by A of frame from B ("A <--B") is at time zero 
(since schedule repeats)
-sending schedule of A: t cells, dividing up the unite line [0,1] in 
intervals of length L1, L2, ..., L{t+1}.
- t>0 (o.w., A is mute).

Observe that any schedule of t cells corresponds 1-1 with a partitioning 
of the unit line [0,1]. Moreover, if (L1,L2, ..., L{t+1}) is a schedule, 
then so is any permutation hereof. As a result, the average size of the 
first segment over all schedules is (L1+ .... + L{t+1})/(t+1)=1/(t+1): 
simply add up the first segment of the schedules (L1,L2, ..., L{t+1}), 
(L2,L3, ..., L{t+1}, L1), ..., (L{t+1},L1, ..., Lt).
The result follows (take t:=C).

Note: this assumes one has infinity many timeslots in a slotframe, so 
that one can approximate rational numbers by real numbers and one can 
ignore Li=Lj occurrences.(

On 2/16/2015 8:14 PM, Malisa Vucinic wrote:
> Hi all,
>
> I prepared a couple of slides on latency aspects with TSCH that can 
> hopefully be useful for evaluating different proposals for the join 
> process once there are more details on message sizes involved.
>
> Regards,
> Malisa
>
>
>
>
>> On 09 Feb 2015, at 20:50, Rene Struik <rstruik.ext@gmail.com 
>> <mailto:rstruik.ext@gmail.com>> wrote:
>>
>> Dear colleagues:
>>
>> Just a quick reminder that we have a 6TiSCH security conf call 
>> tomorrow, Tue February 10, 2015, 4.30pm EST.
>>
>> I have not seen any email traffic the last two weeks, nor any more 
>> feedback on the draft document I posted on Fri January 9th.
>>
>> I put as tentative item on the agenda "time latency aspects" of the 
>> join protocol. I will send out a more definitive agenda tomorrow 
>> around noon EST. Please send agenda request to me prior to that time.
>>
>> Suggested agenda:
>> 1) administrativia {agenda bashing/minutes}
>> 2) time latency aspects
>> 3) AOB
>
>
>
> _______________________________________________
> 6tisch-security mailing list
> 6tisch-security@ietf.org
> https://www.ietf.org/mailman/listinfo/6tisch-security

-- 
email: rstruik.ext@gmail.com | Skype: rstruik
cell: +1 (647) 867-5658 | US: +1 (415) 690-7363